Solve 1-x/4-25x^2-1/5x+2=-7/25x^2+20x+4

$$\frac { 1 - x } { 4 - 25 x ^ { 2 } } - \frac { 1 } { 5 x + 2 } = \frac { - 7 } { 25 x ^ { 2 } + 20 x + 4 }$$

Solve for x

$$x=1$$

$$x=\frac{3}{5}=0.6$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -\frac{2}{5},\frac{2}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5x-2\right)\left(5x+2\right)^{2}, the least common multiple of 4-25x^{2},5x+2,25x^{2}+20x+4.

$$\left(-2-5x\right)\left(1-x\right)-\left(25x^{2}-4\right)=\left(5x-2\right)\left(-7\right)$$

Use the distributive property to multiply -2-5x by 1-x and combine like terms.

$$-2-3x+5x^{2}-\left(25x^{2}-4\right)=\left(5x-2\right)\left(-7\right)$$

To find the opposite of 25x^{2}-4, find the opposite of each term.

$$-2-3x+5x^{2}-25x^{2}+4=\left(5x-2\right)\left(-7\right)$$

Combine 5x^{2} and -25x^{2} to get -20x^{2}.

$$-2-3x-20x^{2}+4=\left(5x-2\right)\left(-7\right)$$

Add -2 and 4 to get 2.

$$2-3x-20x^{2}=\left(5x-2\right)\left(-7\right)$$

Use the distributive property to multiply 5x-2 by -7.

$$2-3x-20x^{2}=-35x+14$$

Add 35x to both sides.

$$2-3x-20x^{2}+35x=14$$

Combine -3x and 35x to get 32x.

$$2+32x-20x^{2}=14$$

Subtract 14 from both sides.

$$2+32x-20x^{2}-14=0$$

Subtract 14 from 2 to get -12.

$$-12+32x-20x^{2}=0$$

Divide both sides by 4.

$$-3+8x-5x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-5x^{2}+8x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=8 ab=-5\left(-3\right)=15$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=5 b=3$$

Rewrite -5x^{2}+8x-3 as \left(-5x^{2}+5x\right)+\left(3x-3\right).

$$\left(-5x^{2}+5x\right)+\left(3x-3\right)$$

Factor out 5x in the first and -3 in the second group.

$$5x\left(-x+1\right)-3\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(5x-3\right)$$

To find equation solutions, solve -x+1=0 and 5x-3=0.

$$x=1 x=\frac{3}{5}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{2}{5},\frac{2}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5x-2\right)\left(5x+2\right)^{2}, the least common multiple of 4-25x^{2},5x+2,25x^{2}+20x+4.

$$\left(-2-5x\right)\left(1-x\right)-\left(25x^{2}-4\right)=\left(5x-2\right)\left(-7\right)$$

Use the distributive property to multiply -2-5x by 1-x and combine like terms.

$$-2-3x+5x^{2}-\left(25x^{2}-4\right)=\left(5x-2\right)\left(-7\right)$$

To find the opposite of 25x^{2}-4, find the opposite of each term.

$$-2-3x+5x^{2}-25x^{2}+4=\left(5x-2\right)\left(-7\right)$$

Combine 5x^{2} and -25x^{2} to get -20x^{2}.

$$-2-3x-20x^{2}+4=\left(5x-2\right)\left(-7\right)$$

Add -2 and 4 to get 2.

$$2-3x-20x^{2}=\left(5x-2\right)\left(-7\right)$$

Use the distributive property to multiply 5x-2 by -7.

$$2-3x-20x^{2}=-35x+14$$

Add 35x to both sides.

$$2-3x-20x^{2}+35x=14$$

Combine -3x and 35x to get 32x.

$$2+32x-20x^{2}=14$$

Subtract 14 from both sides.

$$2+32x-20x^{2}-14=0$$

Subtract 14 from 2 to get -12.

$$-12+32x-20x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-20x^{2}+32x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 32 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-20\right)\left(-12\right)}}{2\left(-20\right)}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-20\right)\left(-12\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-32±\sqrt{1024+80\left(-12\right)}}{2\left(-20\right)}$$

Multiply 80 times -12.

$$x=\frac{-32±\sqrt{1024-960}}{2\left(-20\right)}$$

Add 1024 to -960.

$$x=\frac{-32±\sqrt{64}}{2\left(-20\right)}$$

Take the square root of 64.

$$x=\frac{-32±8}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-32±8}{-40}$$

Now solve the equation x=\frac{-32±8}{-40} when ± is plus. Add -32 to 8.

$$x=-\frac{24}{-40}$$

Reduce the fraction \frac{-24}{-40} to lowest terms by extracting and canceling out 8.

$$x=\frac{3}{5}$$

Now solve the equation x=\frac{-32±8}{-40} when ± is minus. Subtract 8 from -32.

$$x=-\frac{40}{-40}$$

Divide -40 by -40.

$$x=1$$

The equation is now solved.

$$x=\frac{3}{5} x=1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{2}{5},\frac{2}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5x-2\right)\left(5x+2\right)^{2}, the least common multiple of 4-25x^{2},5x+2,25x^{2}+20x+4.

$$\left(-2-5x\right)\left(1-x\right)-\left(25x^{2}-4\right)=\left(5x-2\right)\left(-7\right)$$

Use the distributive property to multiply -2-5x by 1-x and combine like terms.

$$-2-3x+5x^{2}-\left(25x^{2}-4\right)=\left(5x-2\right)\left(-7\right)$$

To find the opposite of 25x^{2}-4, find the opposite of each term.

$$-2-3x+5x^{2}-25x^{2}+4=\left(5x-2\right)\left(-7\right)$$

Combine 5x^{2} and -25x^{2} to get -20x^{2}.

$$-2-3x-20x^{2}+4=\left(5x-2\right)\left(-7\right)$$

Add -2 and 4 to get 2.

$$2-3x-20x^{2}=\left(5x-2\right)\left(-7\right)$$

Use the distributive property to multiply 5x-2 by -7.

$$2-3x-20x^{2}=-35x+14$$

Add 35x to both sides.

$$2-3x-20x^{2}+35x=14$$

Combine -3x and 35x to get 32x.

$$2+32x-20x^{2}=14$$

Subtract 2 from both sides.

$$32x-20x^{2}=14-2$$

Subtract 2 from 14 to get 12.

$$32x-20x^{2}=12$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-20x^{2}+32x=12$$

Divide both sides by -20.

$$\frac{-20x^{2}+32x}{-20}=\frac{12}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{32}{-20}x=\frac{12}{-20}$$

Reduce the fraction \frac{32}{-20} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{8}{5}x=\frac{12}{-20}$$

Reduce the fraction \frac{12}{-20} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{8}{5}x=-\frac{3}{5}$$

Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{3}{5}+\left(-\frac{4}{5}\right)^{2}$$

Square -\frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{3}{5}+\frac{16}{25}$$

Add -\frac{3}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{1}{25}$$

Factor x^{2}-\frac{8}{5}x+\frac{16}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{5}\right)^{2}=\frac{1}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{\frac{1}{25}}$$

Simplify.

$$x-\frac{4}{5}=\frac{1}{5} x-\frac{4}{5}=-\frac{1}{5}$$

Add \frac{4}{5} to both sides of the equation.

$$x=1 x=\frac{3}{5}$$